"...We know that when studying algebraic curves, the choice of ridges and the construction of automorphic forms are key...The key point of the biggest problem is that the Galois representation may only be used in a specific p-adic environment. Effective, when we establish a p-adic number field..

Professor Everton's report continued to penetrate Qiao Yu's ears, but Qiao Yu's mind was completely filled with all kinds of lines at this time. Just a little bit, a little bit...

Not these things.

Because it has been demonstrated in the paper...

The author very carefully used many restrictions to ensure the correctness of the results, and found another way. Wait...

When making mistakes, you should not be bound by existing ideas. As long as it is within the framework established by the other party and does not exceed the framework, he can freely define... an extraordinary singularity, but it is not a common singular structure. ..

"Yes!" Qiao Yu suddenly shouted excitedly.

The conference room suddenly became quiet, and countless eyes focused on the front row.

The professors in the back row could only see the backs of their heads, but the big guys sitting in the first row and Professor Everton on the stage could stare at this man who suddenly shouted and looked excited with confused eyes. juvenile.

After all, this is really rude, even if Qiao Yu is just a minor.

Especially Pan Jingyuan, who was sitting closest to Qiao Yu, was frightened by the sound just now. It was so inexplicable... Qiao Yu also reacted quickly: Fuck, you are in trouble!

So the young man jumped up from his chair like a conditioned reflex, and immediately apologized: "I'm very sorry, Professor Everton, and all the teachers, what Professor Everton just said was really too exciting!

It gave me so much inspiration that I suddenly thought of a very special counterexample, which seemed to prove that a series of papers recently published by Professor Dennis and Professor Sam on ArXiv contained local-global errors in the proof of the Ambidexterity theorem.

This idea made me too excited, and I was so absorbed in thinking just now that I forgot to attend the lecture. I'm sorry, I'm really sorry!" After saying that, Qiao Yu bowed slightly towards Professor Everton on the podium.

After Qiao Yu finished speaking, the conference room became quieter.

Many relevant professors were invited to participate in this series of seminars, but whether it was Yuan Zhengxin or Tian Yanzhen, it was naturally impossible for those who invited these professors to tell the truth. It was because their own students wanted to confirm whether Dennis and Sam had recently Published papers.

After all, that would be too offensive.

I just invited people here in the name of discussing the geometric Langlands conjecture together.

Professors from abroad, domestic, Huaqing, Yanbei and Shuangdan University gathered together to discuss and discuss.

Although everyone knew that after several exchanges, the five papers that had been in the spotlight recently would definitely be involved, but probably no one expected that in the first lecture, the youngest guy in the conference room would jump up and start planning. Destroy the place opposite.

Yuan Zhengxin didn't expect it either, so he also stared at Qiao Yu.

It's a good thing to be inspired, and it's understandable. Even being excited is normal, but there are doubts about the value of the counterexamples in your mind! Well, Professor Everton on the podium naturally didn't expect it, and he glanced at Qiao Yu subconsciously. Pan Jingyuan next to him.

Professor Dennis once introduced this student to him, so he naturally knew him.

Last night, he invited Pan Jingyuan to discuss some ideas about the certification process between him and his mentor.

He couldn't quite figure out why the young kid next to Pan Jingyuan suddenly stood up and said such words. Professor Pan's expression at this time was indeed very strange.

But he didn't look at him, but stared blankly at Qiao Yu standing next to him.

Professor Everton's eyes fell on Qiao Yu, and after thinking for a while, he said: "I can accept your apology, but you have to convince me with what you just thought of. If you can make me think that this idea is indeed great, Not only am I willing to forgive you, I even want to thank you for making this lecture of mine have some different value."

Qiao Yu subconsciously turned his head and glanced at Grandpa Shi, who was sitting in the first row.

Mr. Yuan glared at him first, and then nodded inconspicuously. Qiao Yu was not polite and immediately said: "Okay, Professor Everton, that's what I think. First of all, let's assume...

Everton waved to Qiao Yu and said: "My child, it's not polite to stand down and talk about your thoughts. Come up and speak on the stage. I think your brain definitely didn't prepare the PPT for you, so... .”

As he said that, Everton looked back and smiled: "There happens to be a blackboard here."

The big boss said this, and Grandpa Shi agreed. Qiao Yu, who has never been timid, immediately left his seat and walked to the podium. It happened that things in his mind were a little messy and he could use explanation to clarify his thoughts.

"Teachers, this is what I think. First, suppose a type of singular point of an algebraic variety. Well, this kind of singular point is not very similar to the singular structures we know, such as cusp nodes or cone-shaped singular points. same, with a complex ridge-like expansion globally

The key point is that at the same time, its local geometric structure has a conjugate relationship with other remote points in the algebraic variety, and even non-adjacent singular points. So, first we have to define its local representation. Assume that in A3 or a higher dimension In the geometric space of , its characteristic equation should be..."

With that said, Qiao Yu picked up the chalk and wrote a line of equations on the blackboard: "f(x,y,z)=z^2—x^3y^2 sin(xyz)".

After finishing writing, Qiao Yu took a step back, calculated silently in his heart for a moment, and then continued: "I believe everyone has seen that this equation has a local ridge limit somewhere near the point (0,0,0) structure.

"Well, the conjugate relationship is manifested in the singular points on the algebraic cluster, set as P1 and P2. When they have local ridge singular point structures, their local geometric properties influence each other through a nonlinear homology mapping.

Obviously this means that the local mode structure of the singular point P1 will depend on the local properties of the other remote singular point P2. Note that this conjugate relationship cannot be inferred through simple local geometric observations.

At this point, Qiao Yu's voice suddenly stopped... The audience was also silent, but the reactions were different.

Some people had frowned, picked up pen and paper, and started calculating on the manuscript paper they carried with them; some were still listening carefully; and some were still stunned, watching the development of the situation.

However, Everton on the stage stared at the equation written by Qiao Yu with great interest.

As for Pan Jingyuan in the audience, he was definitely the one with the deepest frown. As the person most familiar with that series of papers, he had vaguely guessed the general idea of ​​​​Qiao Yu, but he had not yet figured out what Qiao Yu would use. Method to break the situation.

But he quickly reacted, glanced at Yuan Zhengxin next to him who was also looking at Qiao Yu seriously, and then took out his mobile phone... The old man might not necessarily show the live video, so he might as well record it himself first.

..

At this time, Qiao Yu's brain was also thinking rapidly.

Although he found the key points, he still had to design an algebraic variety background based on the framework built in a series of five papers.

It happened suddenly. He just had a direction. He had to design this background in a hurry. The test was his ability to perform on the spot. Fortunately, it would probably take some time to calculate the equations he threw out in the audience.

Qiao Yu didn't bother to care what others thought of him now. Anyway, no one was urging him now, so he just thought about it silently.

After thinking like this for five minutes, Qiao Yu, who was standing in front of the blackboard, suddenly picked up the chalk again. This time he did not speak, but started writing directly on the blackboard. "Consider a high-dimensional algebraic variety X, defined as Algebraic variety of the following form: X—1(x,y,z,w)∈A4(z^2—x3y2 sin(xyz) w^5—0)》

After finishing writing, Qiao Yu took a step back again and began to calculate quickly and silently in his brain. It was another minute before he spoke: "Based on the explanation just now, you should be able to find that the algebraic variety X is in (0,0, There is such a singular point near 0,0), and it has a conjugate relationship with the remote point of the algebraic variety through the variable w.

More specifically, P1/P2 are two singular points with the same structure. Yes, that’s right! Then we must consider the ridge structure under the p-adic framework just mentioned by Professor Everton.

As you can see, near the singular points P1 and P2 of So that the mode that appears flat near point P1 no longer maintains projectivity.

Second, because of their conjugate structures, singular points P1 and P2 interact with each other through nonlinear homology algebraic relationships, causing the Ext group to have abnormal behavior near the ridge. That is, the local module structure near P1 cannot be correctly globalized. ization, which undoubtedly destroys the local-global equivalence.

That's all I can think of for the moment. Of course, this is the first step, and it will take some time to analyze the local module structure. Then introduce higher-order category theory and derive the failure of the functor.

If you have read the first paper by Professors Dennis and Sam on the proof of the Langlands Conjecture in geometry, you should know that if it can be proved that the derived functor from local to global cannot satisfy the consistency in the homology category, that is: RHomC(M1,

M2)≈LHomC(M1,M2)

Then it is enough to prove that the local-to-global equivalence in the Ambidexterity theorem fails in this context. The local homology algebraic flatness condition assumed by the theorem is likely to no longer hold when this type of structure exists.

After saying that, Qiao Yu put down the chalk, clapped his hands, and carefully looked at the two formulas he left on the blackboard, then turned around and faced all the professors in the audience and the people standing next to him for the first time.

Professor Everton.

Then Qiao Yu realized belatedly that everyone present, yes, everyone, every one of them, was in an extremely strange state. This included his grandpa, Mr. Yuan, and the people on the stage.

Professor Everton.

It can’t be said that they were dumbstruck. Anyway, everyone’s expressions were different, but they seemed to be frozen.

This reminded him of a very old Japanese TV show. The protagonist on the TV had the ability to stop time, and then only he could move... Of course, what Qiao Yu thought of was "Dinosaur Express"

"No.", but we can't just say that people who think crookedly have unhealthy thoughts. After all, there are too many such movies...

In short, everyone's reaction made Qiao Yu a little uneasy, and then he couldn't help coughing twice: "Ahem, um... Professor Dun, should I go down first?"

"...Okay...you can go down first, kid. By the way, if I guess correctly, you are Professor Yuan's grandson. Your name is Qiao Yu, right?"

Professor Everton came back to his senses and asked.

"Yes, Professor Everton." Qiao Yu, who had just stepped off the podium, turned around and answered. "Sixteen years old?" Everton asked again.

Qiao Yu nodded.

Professor Everton pursed his lips, glanced in the direction of Yuan Zhengxin, then gave a thumbs up and said, "You are great!"

Then watching Qiao Yu sit back in his seat, he glanced at his PPT and continued: "Then let's continue... Forget it, let's first roughly calculate what Professor Yuan's grandson Qiao Yu just did.

The counterexamples we gave..."

As he spoke, Everton simply turned around, picked up the chalk, and began to perform some simple deductions on the black plate.

After a moment, Everton suddenly turned around and asked: "Qiao Yu, this is a type of singularity you defined for the first time. Maybe you should give it a name, otherwise I don't know how to express it."

"Uh..." Qiao Yu obviously hadn't thought about this problem at all and was stunned.

"Well, it seems you haven't thought about it yet, so let's call him JOE." Everton shrugged, then turned his head and continued to deduce on the blackboard.

Qiao Yu took a deep breath and didn't look at the old professor's deduction process. He had already calculated it in his mind and it should be fine. So he just glanced sideways at Professor Pan Jingyuan beside him.

Speaking of which, the person he felt most sorry for for doing this was Professor Pan.

He had patiently guided him for so long, and he turned around to prove that the paper he had been involved in researching for many years was flawed, and he was somewhat embarrassed. Who knew that when he secretly looked at him, Professor Pan turned to look at him openly, looking at each other, Qiao. Yu felt even more embarrassed.

He smiled coquettishly and whispered, "Well, Professor Pan, please don't be depressed first. What I'm thinking may not be right."

"Hmm..." Jingyuan responded with a soft voice. Qiao Yu couldn't figure it out and didn't know what it meant, but his sitting posture and expression were in sync with each other, and there was no hint of any embarrassment.

Again, no matter how much armor the instructor and grandpa Shi add to him, the increased armor will be magic defense, but it cannot prevent physical attacks. Brother Pan is right next to him, so he must be more honest in this situation.

It is a well-known fact that it is terrible for an honest person to get angry.

Professor Everton stood on the podium silently and calculated that the time had exceeded the originally set time. Gradually, some people in the audience thought of something and stood up and walked to the podium to discuss with Professor Everton. , so there were more contents written in different handwriting on the blackboard.

With the first, there is the second...

Soon, the podium was surrounded. In the process, even Pan Jingyuan next to him stepped onto the podium. Qiao Yu was stunned...

What are you doing? Is this lecture finished? Can he leave? So Qiao Yu looked at Mr. Yuan on the other side.

The old man was smiling and watching the discussion on the stage. Then he seemed to realize something and turned to look at Qiao Yu, who was fidgeting. He probably guessed what Qiao Yu was thinking, so he waved slightly to Qiao Yu.
To be continued...